Magnetic backgrounds and tachyonic instabilities in closed string theory
Cite as: AIP Conference Proceedings 607, 269 (2002); https://doi.org/10.1063/1.1454381 Published Online: 25 February 2002
A. A. Tseytlin
AIP Conference Proceedings 607, 269 (2002); https://doi.org/10.1063/1.1454381 607, 269
© 2002 American Institute of Physics. Magnetic Backgrounds and Tachyonic Instabilities in Closed String Theory
A.A. Tseytlin
Department of Physics, The Ohio State University, Columbus OH 43210-1106, USA
Abstract. We consider closed superstrings in Melvin-type magnetic backgrounds. In the case of NS-NS backgrounds, these are exactly solvable as weakly coupled string models with spectrum containing tachyonic modes. Magnetic field allows one to interpolate between free superstring the- ories with periodic and antiperiodic boundary conditions for the fermions around some compact direction, and, in particular, between type 0 and type II string theories. Using "9-11" flip, this inter- polation can be extended to M-theory and may be used to study the issue of tachyon condensation in type 0 string theory. We review related duality proposals, and, in particular, suggest a description of type 0 theory in terms of M-theory in a curved magnetic flux background in which the type 0 tachyon appears to correspond to a state in d=l 1 supergravity fluctuation spectrum.
INTRODUCTION
Magnetic backgrounds play an important role in field theory and open string theory pro- viding simple solvable models with nontrivial physics content. Similar (approximately) constant magnetic backgrounds in gravitational theories like closed string theory are necessarily curved with an example provided by the Melvin-type flux tube solutions (see, e.g., [1]). The role of the magnetic field(s) is played by the vector(s) originating from the metric and/or antisymmetric 2-tensor upon Kaluza-Klein reduction on a circle. One important special case is the Kaluza-Klein Melvin background which has curved magnetic universe interpretation in 9 dimensions, but, remarkably, is represented by a flat ("twisted") 10-dimensional space [2]. The non-trivial 3-dimensional part of this space is a twisted product of a K-K circle and a 2-plane: going around the circle must be accompanied by a rotation by angle 2nb in the plane. This is a continuous version of the Rohm model [3] where the 2-plane was replaced by a 2-torus so that the magnetic ("twist") parameter was allowed to take only discreet values. The flatness of the 10-d space allows one to solve the corresponding (super)string theory exactly [4, 5] determining the "Landau-type" spectrum of states which (for large enough magnetic field) contains tachyons in the winding sector. Other NS-NS Melvin- type models (which are no longer flat in 10 dimensions) are also solvable since they may be formally related to the K-K Melvin model by a generalized O(d,d) T-duality transformation [6]. One can also construct similar magnetic backgrounds with R-R magnetic field by applying U-duality transformation, or equivalently, by lifting, e.g., the K-K Melvin space to 11 dimensions and then reducing down to 10 along a different dimension [7, 8]. Such R-R string models seem, however, hard to solve exactly.
CP607, String Theory, edited by H. Aoki and T. Tada © 2002 American Institute of Physics 0-7354-0051-2/027$ 19.00 269 One characteristic feature of these magnetic closed string models is that the magnetic field parameter introduces an effective phase for space-time fermions; in particular, switching on magnetic field allows one to interpolate between periodic and antiperiodic boundary conditions for the fermions along a spatial circle [4, 7]. Antiperiodic fermions appear in the context of superstrings at finite temperature [9] and in closely related type 0 string theory viewed as (—I)FS orbifold of type II superstring theory [10, 11]. Using the orbifold interpretation of type 0 string theory and "9-11" flip it was sug- gested in [12] to interpret type 0 string theory as a similar non-super symmetric orbifold compactification of 11-d M-theory. It was conjectured [12] that the tachyon of type OA string in flat space gets m2 > 0 at strong coupling where type OA string becomes dual to 11-d M-theory on large circle with antiperiodic fermions. In an interesting paper [13], this proposal was combined with the observation that K-K Melvin magnetic background in type II string theory (and its direct lift to 11 dimensions) may be used to interpolate [4, 7] between periodic and antiperiodic boundary conditions for space-time fermions. Since reducing the flat twisted 11-d space along a different - "mixed circle" - direction gives [2, 7] the R-R 7-brane background, it was suggested that type 0 string in this 10-d R-R Melvin background should be dual to type II theory in the same background but with shifted magnetic field. This implies that type 0 string tachyon should disappear at strong magnetic R-R field [13, 14]. In our recent work [8], which will be reviewed below, we extended the discussion in [7, 13] to the case of more general class of Melvin-type backgrounds with two independent magnetic parameters b and b (parametrizing the vector and axial vector 9- d fields originating from G® and B& components). This class includes the K-K Melvin (b ^ 0, b = 0) and the dilatonic Melvin (b — b) solutions as special cases and is covariant under T-duality. We lifted these type IIA magnetic solutions to 11 dimensions, getting a b ^ 0 generalization of the flat [2] d=l 1 background discussed in [7, 13]. Dimensional reduction along different directions (or U-dualities directly in d=10) led to a number of d=10 supergravity backgrounds with R-R magnetic fluxes, generalizing to b ^ 0 the R-R magnetic flux 7-brane [7, 13] of type IIA theory. We proposed the analog of the type II - type 0 duality in [13] now based on a curved d=l 1 background which is a lift of the T-dual b ^ 0, b — 0 Melvin background. Remark- ably, this background has manifest 9-11 symmetry, i.e. its two different 10-d reductions are formally the same NS-NS Melvin spaces. While in the original K-K Melvin set-up the tachyon originates from a stringy winding mode (i.e. type 0 tachyon may be inter- preted as corresponding to a wound membrane state) in this T-dual setting the tachyon appears to be a momentum mode, i.e. the type 0 tachyon appears to correspond to a state in d=l 1 supergravity multiple!. One motivation for the study of these magnetic backgrounds and their instabilities is to get better understanding of closed string tachyons and their stabilization, i.e. of dynamical interpolations between stable and unstable theories. In particular, one would like to establish precise relations between non-supersymmetric backgrounds in type II superstring theory and unstable backgrounds in type 0 theory. Another is to use interpolating magnetic backgrounds to connect D-brane solutions in the two theories and related gauge theories. Understanding the fate of closed string tachyon in such non- supersymmetric situations is crucial, in particular, for the program of constructing string theory duals of non-supersymmetric gauge theories [15, 16] (see, e.g., [17, 18, 19] for
270 some related work).
NS-NS MELVIN BACKGROUND IN D^IO SUPERSTRING
Consider the following NS-NS bosonic background in type II string theory: r2 (1) + bzr2
This is an exact solution of the string theory - the corresponding sigma model is 7 conformal to all orders in a [4, 5], x9 is a periodic coordinate of radius R = R9 and
type flux tube interpretation becomes apparent upon dimensional reduction in the x9- direction. The resulting solution of d = 9 supergravity (3) (4) (5) describes a magnetic flux tube universe with the magnetic vector A (coming from the metric) and the axial-vector B (coming from the 2-form) and the non-constant dilaton <|) and K-K scalar a. There are two special important cases: (i) b = Q,b ^ 0 and (ii) b = 0,£ ^ 0. The corresponding string backgrounds are T-dual to each other. The first (K-K Melvin model) is represented by the flat 10-d space (but curved magnetic 9-d background)
2 2 2 2 2 ds = -dt + dx] + dxl + dr + r (d($> + b dx9) . (6) The second has non-trivial metric, 3-form and dilaton:
2 2 2 2 l 2 2 2 ds w = -dt + dx + dr + f~ (dx 9 + r
2 H3 = dB2 = 2bf~ rdr A dy A dx9 , ^(^-^o) = f- 1 _ (8) They define equivalent conformal field theories, with the two different geometries being "seen", as in other known T-dual situations, by point-like momentum and winding string modes respectively. *
1 While the curved 2-parameter background (1) may look much more complicated than the flat one (6), it is the natural T-duality (O(2,2) duality) "completion" of (6). A possible analogy is to think of (6) as a counterpart of a plane wave background; then (7) corresponds to the fundamental string background, and (1) - to a superposition of a fundamental string and a wave.
271 For generic values of b and b all of the type II supersymmetries are broken; supersym- metries are restored at the special values bR = 2n\ and bR — 2ri2 (HI = 0, ± 1,...), where the conformal model describes the standard flat-space type II superstring theory. As follows from (6), a shift of x$ by period of the circle 2nR implies rotation in the plane by angle 2nbR. Thus in the special case of bR = 2k + 1 (k — 0, ± 1,...) the metric becomes topologically trivial. However, the superstring theory in this background is still non-trivial (not equivalent to that of standard flat space) since space-time fermions change sign under 2n rotation in the plane. That means that the bR = 1 case (all models with bR = 2k 4- 1 are equivalent) describes superstrings with anft'periodic boundary condition in xg, just as in the twisted 3-torus model of [3] or in the finite temperature case [9]. Type IIA string in the space (6) with bR = 1 is equivalent, for R —> 0, to type l l OA string in R >* x SJ^0 or type OB string in R >* x SJ^^. The tachyon of type 0 theory originates from a particular winding mode present in the bR=l type II model spectrum. The string model corresponding to the above 2-parameter background can be solved in terms of free fields and its spectrum is found to be
l 2 f 2 2 l 2 -v!M = ±a (E -p )=NR + NL + -RR-\m-bRJ]
l 1 2 + -RR~ (w - bRJ] - j(JR - JL) = 0 , (9) where NR — NL — ww, and y = bRw + bRm — bRbRJ . Here ps are continuous momenta in the 6 free directions, the integers m and w represent quantized momentum and winding numbers in the compact xg direction, $R and RL are the number of states operators, which have the standard free string theory form in terms of normal-ordered bilinears of bosonic and fermionic oscillators, and / = JL + JR and f^R are the angular momentum operators in the 2-plane with the eigenvalues SL,R — i(t,/? + 5) + SL,R, / = IL — IR + SL + SR. The orbital momenta IL,R = 0,1,2,... (which replace the continuous linear momenta p\, pi in the (r, cp) 2-plane for non-zero values of 7) are the analogues of the orbital quantum number / and the radial quantum number k in the Landau problem, and SR,L are the spin components. The spectrum is thus similar to the one for charge particles in constant magnetic field orthogonal to the plane of motion (with Q^R = mR~l ± w/?"1 playing the role of charges of string states in 9-dimensional description). In contrast to the open string case, here not only the masses and spins but also charges of string states can take arbitrarily large values. The terms in M2 that are linear in angular momenta reflect gyromagnetic interaction, while the terms quadratic in /'s represent the effect of gravitational back reaction of the magnetic field. Since / can take both integer (in NS-NS, R-R sectors) and half-integer (in NS- R, R-NS sectors) values, the symmetry of the bosonic part of the spectrum b —> b + k\R~l, b —>• b + k2&~1 is not a symmetry of its fermionic part, i.e. the full superstring spectrum is invariant under the shifts with even integer kt — 2nt only. In the case of bR = 2n\, bR = 2ri2 (i.e. y =even integer, y = 0) the spectrum is thus equivalent to that of the standard free superstring compactified on a circle. In the two cases (a) bR = odd, bR = even or (b) bR = even, bR = odd the spectrum is the same as that of
272 the free superstring compactified on a circle with awft'periodic boundary conditions for space- time fermions. The T-duality symmetry in the compact Kaluza-Klein direction x9 is manifest in the spectrum: M2 is invariant under R R = a'R~1^ b focr. The spectrum of the b = 0 model (y = bRw) has the first (lowest mass) potentially tachyonic state that appears as b is increased from zero for /= 0, JR — JL = 1: it has NR = NL = 0, m = 0, w = 1, 1R = 1L = 0, SR = -SL = 1 , and thus 2 a'M = -lbR = 2R(bcr-b) , bcr = , i.e. we get tachyon in the winding sector when b > bcr. In the T-dual model with b = 0, b ^ 0, which has the equivalent spectrum, the first tachyon appears in momentum sector for b>bcr=.j = . RELATED D=ll BACKGROUNDS AND U-DUAL D=10 R-R FLUX BRANES Lifting the type IIA background (1) to d=ll one gets the following "6-brane" (or "8- brane" with "defects" along jcp, x\ 1) solution of d=l 1 supergravity [8] ^ , (10) This background is regular, with curvature being constant near the core r = 0 and going to zero at large r. The two special cases are b = 0 when the d=l 1 space becomes flat (this is the direct d= 11 lift of (6)) , C3 = 0 , (11) and b = 0 when the metric and CB are still non-trivial 2 l 2 2 2 2 2 2 2 2 ds n =f f\-dt + dx }+f- f\fdr + r d C3 = 273 Remarkably, this metric and \C^\ turn out to be symmetric under the "9-11" flip. The reduction of (10) along jcn gives back (1), while the reduction along x$ leads to the following "mixed"(NS-NS/R-R) type IIA background related to (1) by a U-duality (Tj^ST^ duality transformation) A = brrd • x$. The two parameters b and b now control the strengths of the magnetic 1-form R-R field A and of the NS-NS 2-form field #2- This solution may be interpreted as a R-R magnetic flux 7-brane with a "defect" along JCQ. In the special case of b — 0 we get the R-R flux 7-brane background [7] which is the reduction of the flat d=l 1 background (11) along the other -x$ - direction (and thus is U-dual of the flat K-K Melvin space (6) which is the trivial reduction along x\\) d Like in the NS-NS Melvin case (4), the strength of the the R-R 1-form is approximately constant near the core, decaying away from the core. However, in contrast to (2) here the dilaton grows away from the core, so the theory is weakly-coupled only at small r<^b~l [7,13]. In the other special case b = 0 the 9 — 11 symmetry of (12) implies that here the two "U-dual" reductions are formally equivalent, leading to the same (pure NS-NS) background (7) (with the dilaton decreasing away from the core). PERTURBATIVE AND NON-PERTURBATIVE TYPE 0 - TYPE II RELATIONS The above considerations suggest [13, 8] non-perturbative dualities relations between type II and type 0 strings in magnetic backgrounds. Let us first review the well-known perturbative flat space orbifold relation between these two string theories. Type 0 closed string theory [10] which is a "symmetric" ( — I)FL+FR modular-invariant FS projection of the NSR string may be interpreted also as (— 1 ) (Fs is space-time fermion number) orbifold of type II superstring theory. The orbifolding projects out all space- time fermions and introduces extra twisted sector states (in particular, the tachyon and another copy of massless R-R bosons). This relation between type II and type 0 string theories may be expressed by saying that type 0 string is (a limit of) type II string com- pactified on a circle with antiperiodic boundary conditions for space-time fermions [9]. More precisely, the two theories are the limits of the same interpolating "9-dimensional" l FS string theory [11] - I,R orbifold of type II theory. I/? stands for (S )/?/[(-I} x S], 274 where S is half-shift along the circle (X$ —> X$ + nR).2 Type IIA on S/^oo is type IIA theory in flat d=10 and type IIA on ^R^Q is type OA theory on (SI)R-+Q (or T-dual type OB theory on (Sl )/?-»«>). Thus the E/? orbifold type IIA theory (which has massive fermions in its spectrum) continuously interpolates between supersymmetric type IIA and non-super symmetric type OB ten-dimensional theories [11]. It is possible to replace the orbifolding procedure by the K-K Melvin background with bR—\. Indeed, using that type II superstring in the d=10 K-K Melvin NS-NS background (6) (with x$ = x$ + 2nR) is, for bR = 1, equivalent to type II theory on R9 x (S1 )R with antiperiodic boundary conditions for the space-time fermions along the JCQ circle [4],3 one may notice [13] that it is equivalent to the above orbifold of type II on Z/?/ with the radius Rf = 2R (extra 2 is related to the shift S).4 This NS-NS Melvin model thus describes, in the limit R —> 0, the weakly coupled type 0 string on Rl>s x (SI)R->Q (or type OB on fl1'8 x (S1)/^,). To summarize, the type II orbifold on IQR is the same as type II theory in Melvin background at bR = 1 and thus the latter model interpolates between type II and type 0 theories in infinite d=10 space. In general [13], type IIA(B) theory in the Kaluza- Klein NS-NS Melvin background with parameters (b,R) is equivalent (has the same perturbative spectrum and thus the same torus partition function) to the orbifold of type IIA(B) on Z/e/ in this NS-NS Melvin background with parameters (b1 = b — R~l, Rf = 2R). Since the K-K Melvin model (6) has the same spectrum as the T-dual model (7), a similar statement is true also for the type II and type 0 strings in this dual curved background [8]. Returning now back to the case of the trivial flat background, the above com- 1 pactification of type IIA theory on Z/? = Z/?9 corresponds to M-theory on (S )Ri x l Fs (S )R9/[(-l) x S], where R9 and R'n are taken to be small. Making the "9-11" flip (i.e. exchanging the roles of the two circles, assuming that there is indeed an interpolat- ing coordinate-invariant 11-d M-theory) one may then conjecture [12] that one should get an equivalent description of this theory as an ordinary (i.e. with periodic fermions) S1 compactification of the d=10 theory obtained from M-theory on E^/ (in the limit R'u —> 0). Then type OA theory may be interpreted as M-theory on ER/ , or, essentially, 1 as M-theory on S with radius ^R\ { with periodic boundary conditions for the bosons and antiperiodic for the fermions. Lifting the flat K-K Melvin metric (6) to 11 dimensions, replacing x$ 2 The role of S is to mix (— 1)F* orbifold with compactification on the circle, allowing for the interpolation [11]. The action of S is irrelevant for R-^o°, but is crucial for reproducing type 0 spectrum for R —> 0. 3 This model atbR= 1 is essentially the same as a limit (T2 —>• R2) of the twisted 3-torus T2 x S1 model of [3] or a Wick rotation of the finite temperature superstring theory [9], and is closely related to Scherk- Schwarz-type compactification in string theory [20]. 4 In the Melvin model with bR = 1 the shift xg -»> xg + 2nR is equivalent to 2n rotation in the 2-plane under which space-time fermions change sign. In the orbifold, the corresponding shift S is by nR1. 275 7-brane background (14)5 are non-perturbatively dual to each other, with parameters 1 related by b0 = b - fl" , gs0 = ^ = ^JJL. For example, for bRn = 1 the d=l 1 Melvin theory has antiperiodic fermions on the circle R\ \, while, according to [12], type 0 theory is M-theory with antiperiodic fermions on circle \R'n» That means, in particular, that starting with type 0 theory and increasing the value of the R-R magnetic parameter bo, the tachyon should disappear at strong enough magnetic field [13] when the theory becomes strongly coupled - its weakly-coupled description is in terms of stable weakly coupled type II theory in R-R Melvin background with small b.6 Replacing the starting point of the above discussion - K-K Melvin space (6) (and thus also its d=l 1 counterpart (11)) by the perturbatively equivalent T-dual background (7) (with the associated d=ll background (12)) we are able to give an alternative formulation [8] of the type II - type 0 duality conjecture. Indeed, there are two magnetic type II models which are equivalent to type II superstring theory in flat space with fermions obeying antiperiodic boundary conditions in the JCQ direction: (a) the model with b = 0, bR = 1 ; (b) the model with bR = 1, b = 0. They are equivalent (T- dual) as weakly-coupled type II superstring theories compactified on a circle. T-duality equivalence is believed to hold also for finite coupling, so one expects that M-theory compactified along JCQ,x\\ in the background (11) with bR = I and in (12) with bR = 1 should be equivalent not only in the weak-coupling limit R\\ —> 0 but also for finite J?n. Thus it is natural to conjecture that M-theory in the background (12) with bR$ = 1 compactified on the (jC9,jtn) torus with (+,+) (periodic) boundary conditions for the fermions is equivalent to M-theory in flat space compactified on the 2-torus with the (—, +) fermionic boundary conditions. One may thus propose the following description of type OA string theory: as M-theory in the background (12) with periodic xg and bRg = 1. Because of the 9-11 symmetry of (12) one may consider also the equivalent model within = 1. While the reduction of (11) to 10 dimensions gives string theory in R-R F7 back- ground, the reduction of (12) gives apparently much simpler string theory in the NS-NS background (7) the spectrum of which (in the weakly-coupled regime) we know. How- ever, the 9-11 flip necessary to relate the type II and type 0 theories implies that, e.g., a weakly coupled (Ru^R$) type II theory is mapped into a strongly coupled (R$ 5 Any bosonic solution of type II supergravity can be embedded into type 0 theory provided the fields of the twisted sector (tachyon and second set of R-R bosons) are set equal to zero [16]. 6 As explained in detail in [8], one indication that, as in the weakly-coupled AdS$ x S5 example in [16], 2 the R-R flux may shift the value of the tachyon mass follows from the presence of a non-minimal T F%n coupling [16] of the tachyon to the R-R background. 7 In particular, the weakly coupled type 0 theory in the NS-NS Melvin background is certainly unstable for any value of magnetic field parameter (because of its own tachyon present already in flat space) but it is expected to become stable at strong coupling and critical magnetic field since it should then become equivalent to a weakly coupled type II string in NS-NS background with a small magnetic field parameter. 276 Melvin background (14) are equivalent, being related by a shift of the R-R field strength parameter: bo = b — R^ . More generally, the discussion in [8] implies (assuming again the validity of the 9-11 flip) the following generalization of this relation to the case of the 2-parameter magnetic Melvin model (R — R\\)\ Type IIA in (b,b) Melvin = Type OA in (b - R~ l , b) Melvin = Type OA in (b,b - R~ l ) Melvin. It was conjectured in [12] that the type OA tachyon originates from a massive mode (in twisted sector of Z-orbifold) in microscopic M-theory, i.e. a mode which is absent in the d=l 1 supergravity spectrum. The magnetic instability (related to type 0 tachyon) appears in different ways in the two models. In the first case the tachyon is associated with a winding string mode (or winding membrane state in M-theory, see [8]). In the second M-theory description of type OA theory - as M-theory in the background (12) at the critical magnetic field bRg = 1 the type OA tachyon corresponds to an unstable mode in the momentum part of the spectrum which may be thus seen directly in d=10 or d=l 1 supergravity spectra in the corresponding curved backgrounds (7) and (12).8 This setting thus allows one to interpret the type OA tachyon as a particular fluctuation mode of d=l 1 supergravity expanded near (12). The solution of the relevant Laplace operatore has the following structure [8] (15) where J = IL — IR + SL + SR and 779 = m/Rg, p\\ — n/R\\ (R$ > R\\). The tachyon appears for IL — IR = 0, SR = — SL = 1, w = ±1, n = Q. The corresponding tachyonic mode in the "dual" description of type 0 theory as M-theory on the flat background (11) may be interpreted as a particular wound membrane state [8] 2 2 2 2 M = (^wRgRi i T2) + b J + ZT^bwRgRi i T2(1L + IR + 1 - SR + SL) , (16) where T% = (4n2Ri\o!)~l. The first term represents the mass of a membrane of tension T2 wrapped on a 2-torus (wound w times around xg and once around Jtn). The spectrum is similar to (15), as expected from the T-duality relation between the corresponding ten-dimensional models. The counterpart of the first two terms in (15) is now a winding term; the gyromagnetic interaction in (15) is traded for an analogous gyromagnetic term where the charge is now the winding number w.In this way we determine part of the spectrum for the d=ll flat model (11) which is relevant for the study of instabilities. The winding membrane state that becomes tachyonic at sufficiently large b has quantum numbers SR= 1 = —Si, IL = !R = 0. ACKNOWLEDGMENTS I would like to thank the organizers for the excellent conference and hospitality. The work described in this contribution was done in collaboration with J. Russo [8] and was 8 While K-K Melvin background is stable as a supergravity solution (the string instability is associated with a winding mode), the curved T-dual background (7) is unstable already at the supergravity level. 277 partially supported by the DOE grant DE-FG02-91ER40690 and also by the INTAS 99-1590 and CRDF RPI-2108 grants. REFERENCES 1. G.W. Gibbons, "Quantized Flux Tubes In Einstein-Maxwell Theory And Noncompact Internal Spaces," in: Fields and Geometry, Proceedings of the 22nd Karpacz Winter School of Theoretical Physics, ed. A. Jadczyk (World Scientific, Singapore, 1986). 2. F. Dowker, J. P. Gauntlett, S. B. Giddings and G. T. Horowitz, "On pair creation of extremal black holes and Kaluza-Klein monopoles," Phys. Rev. D 50, 2662 (1994) [hep-th/9312172]. F. Dowker, J. P. Gauntlett, G. W. Gibbons and G. T. Horowitz, "The Decay of magnetic fields in Kaluza-Klein theory," Phys. Rev. D 52, 6929 (1995) [hep-th/9507143]. 3. R. Rohm, "Spontaneous Supersymmetry Breaking In Supersymmetric String Theories," Nucl. Phys. B 237, 553 (1984). 4. J. G. Russo and A. A. Tseytlin, "Magnetic flux tube models in superstring theory," Nucl. Phys. B 461, 131 (1996) [hep-th/9508068]. 5. A. A. Tseytlin, "Closed superstrings in magnetic field: instabilities and supersymmetry breaking", Nucl. Phys. Proc. Suppl. 49, 338 (1996) [hep-th/9510041]. 6. J. G. Russo and A. A. Tseytlin, "Exactly solvable string models of curved space-time backgrounds," Nucl. Phys. B 449, 91 (1995) [hep-th/9502038]. A. A. Tseytlin, "Exact solutions of closed string theory," Class. Quant. Grav. 12, 2365 (1995) [hep-th/9505052]. 7. J. G. Russo and A. A. Tseytlin, "Green-Schwarz superstring action in a curved magnetic Ramond- Ramond background," JHEP9804, 014 (1998) [hep-th/9804076]. 8. J. G. Russo and A. A. Tseytlin, "Magnetic backgrounds and tachyonic instabilities in closed super- string theory and M-theory," hep-th/0104238. 9. J. J. Atick and E. Witten, "The Hagedorn Transition And The Number Of Degrees Of Freedom Of String Theory," Nucl. Phys. B 310, 291 (1988). 10. L. J. Dixon and J. A. Harvey, "String Theories In Ten-Dimensions Without Space-Time Supersym- metry," Nucl. Phys. B 274, 93 (1986). N. Seiberg and E. Witten, "Spin Structures In String Theory," Nucl. Phys. B 276, 272 (1986). 11. J. D. Blum and K. R. Dienes, "Strong/weak coupling duality relations for non-supersymmetric string theories," Nucl. Phys. B 516, 83 (1998) [hep-th/9707160]. M. Dine, P. Huet and N. Seiberg, "Large And Small Radius In String Theory," Nucl. Phys. B 322, 301 (1989). 12. O. Bergman and M. R. Gaberdiel, "Dualities of type 0 strings," JHEP9907, 022 (1999) [hep- th/9906055]. 13. M. S. Costa and M. Gutperle, "The Kaluza-Klein Melvin solution in M-theory," JHEP 0103, 027 (2001) [hep-th/0012072]. 14. M. Gutperle and A. Strominger, "Fluxbranes in string theory," JHEP 0106, 035 (2001) [hep- th/0104136]. 15. A.M. Polyakov, "String theory as a universal language," hep-th/0006132. "The wall of the cave," Int. J. Mod. Phys. A14 (1999) 645 [hep-th/9809057]. 16. I. R. Klebanov and A. A. Tseytlin, "D-branes and dual gauge theories in type 0 strings," Nucl. Phys. B 546,155 (1999) [hep-th/9811035]. "A non-supersymmetric large N CFT from type 0 string theory," JHEP 9903, 015 (1999) [hep-th/9901101]. 17. I. Antoniadis, J. P. Derendinger and C. Kounnas, "Non-perturbative temperature instabilities in N = 4 strings," Nucl. Phys. B 551, 41 (1999) [hep-th/9902032]. 18. A. Adams and E. Silverstein, "Closed string tachyons, AdS/CFT, and large N QCD " hep-th/0103220. 19. A. Adams, J. Polchinski and E. Silverstein, "Don't panic! Closed string tachyons in ALE space- times," hep-th/0108075. 20. C. Kounnas and B. Rostand, "Coordinate Dependent Compactifications And Discrete Symmetries," Nucl. Phys. B 341, 641 (1990). 278